The present invention relates to a method of spectrum analysis in two-dimensional representation to obtain accurate characteristic information on an object to be analyzed by using derivatives on spectral data of the said object in spectroscopic analysis.
Heretofore, in spectroscopic analysis (including infrared spectrophotometry and the like) in general, by measuring an intensity of light transmitting through the object to be analyzed(may hereafter be denoted as xe2x80x9canalyzed objectxe2x80x9d) and converting absorbance, various characteristics or information in terms of physical and/or chemical properties (may hereafter be denoted as xe2x80x9ccharacteristic informationxe2x80x9d) including shapes, concentration, and the like of substances contained in the said analyzed object have been obtained based on the shapes of the said spectrum or a spectral profile. As such a conventional spectral profile, that spectral profile representing absorbance of the analyzed object in terms of the wavenumber (or wavelength) are generally used (may hereafter be referred to as xe2x80x9cabsorbance/wavenumber spectral profilexe2x80x9d). It can be considered that such a spectral profile is formed with a number of component bands overlapped.
As a means to analyze such spectral profiles, for example, the Japanese patent publication H11-148865 discloses the method of using derivatives on the spectral waveform of the analyzed object in spectroscopic analysis. To be concrete, as for the specific analyzed object, when spectral waveform where the intensity of the signal output of a spectrophotometer is represented as a function of wavenumber, wavelength, or time is prepared letting n1 and n2 (n1xe2x89xa0n2) be a positive integer, a method of spectrum analysis in two-dimensional representation with the following characteristics is disclosed. These characteristics are;
Calculating the n1-th and n2-th derivatives with respect to wavenumber, wavelength, or time of the said intensity of the signal output regarding the spectral profile on the said spectral data, plotting points on the two-dimensional coordinate plane whose X-coordinate is the said n1-th derivative and whose Y-coordinate is the said n2-th derivative, respectively, on the said two-dimensional coordinate plane, creating a two-dimensional plot of derivative pairs on the said spectral waveform, thereby obtaining specific information on the said spectral profile based on the two-dimensional plot of the said derivative pairs.
However, although the spectral analysis method cited in the Japanese patent publication H11-148865 can find component bands when the spectral profile contains a (single) component band, it is difficult to find all these component bands and even to estimate them is extremely difficult when the spectral profile of the analyzed object contains several component bands overlapped. In particular, there are many cases that the spectral profile of the analyzed object has several overlapping component bands. Therefore, as for the analyzed object with a spectral profile containing several component bands, the method described in the Japanese patent publication H11-148865 can estimate the dominant component bands to some extent among the said component bands. However, as for unresolved bands buried under the dominant band, that is, the characteristics of those component bands are unclear due to severe overlap, it is extremely difficult even to estimate those bands.
Based on the method cited in the Japanese patent publication H11-148865, the present invention has been developed for further improvement. As a result of intensive studies, as for the specific analyzed object, when spectral data where the intensity of the signal output of a spectrophotometer is represented as a function of wavenumber, wavelength, or time, are prepared letting n and m (nwm) be a positive integer, the inventor has proposed a method of spectrum analysis in two-dimensional representation for obtaining the specific characteristic information on the said spectral data based on the said two-dimensional derivative plot by calculating the n-th and m-th derivatives with respect to wavenumber, wavelength, or time, of the said spectral data, plotting points on the two-dimensional coordinate plane as the X-Y coordinate system whose X-coordinate is the said n-th derivative and whose Y-coordinate is the said m-th derivative respectively on the said two-dimensional coordinate plane, and preparing the two-dimensional derivative plot on the said spectral data.
Further, based on the characteristic information described above, the inventor has proposed a method of spectrum analysis in two-dimensional representation which estimates the component bands comprising the spectral profile of the analyzed object by estimating band parameter values regarding at least one component band among the component bands contained in the spectral profile of the analyzed object, estimating at least one component band, obtaining the two-dimensional derivative plot with the specific remaining component bands removed by clearing the specific component band or specific component bands already estimated or the two-dimensional derivative plot from spectral profiles or the two-dimensional derivative plot of analyzed object, obtaining specific characteristic information based on the two-dimensional derivative plot of this specific component removed, estimating band parameter values on other component bands based on the said characteristic information, and iterating the estimation of at least one of the other component bands thereby estimating component bands in order.
That is, the present invention is a method of estimating component bands in order, using a specific component band or specific component bands already estimated and estimating remaining component bands (including the ones not yet estimated, the ones desired to be optimal).
To be concrete, a preferred embodiment of the present invention, although not specifically limited, is, first of all, to estimate one component band BDi(i is a positive integer) by estimating the band parameter values for BDi among component bands contained in the spectral profile of the analyzed object based on the specific characteristic information on the spectral profile of the two-dimensional derivative plot.
Next, the preferred embodiment is to prepare the two-dimensional derivative plot with BDi removed, either creating the two-dimensional derivative plot with BDi removed by clearing the two-dimensional derivative plot of the component band BDj (j is a positive integer where jxe2x89xa0i) from the two-dimensional derivative plot of the said analyzed object or obtaining the spectral profile with BDi removed by clearing a profile of BDi from a spectral profile of the said analyzed object.
Then, the preferred embodiment is to estimate one component band BDj by estimating band parameter values on BDj other than BDi among the component bands contained in the spectral profile of the analyzed object based on the said characteristic information by obtaining the specific characteristic information based on the said two-dimensional derivative plot with BDi removed.
Next, the preferred embodiment is to prepare a two-dimensional derivative plot with BDj removed, either creating the two-dimensional derivative plot with at least BDj removed by clearing BDj or the two-dimensional derivative plot of both BDi and BDj from the two-dimensional derivative plot of the said analyzed object or obtaining the spectral profile with at least BDj removed by clearing BDj or the profile of both BDi and BDj from a spectral profile of the said analyzed object.
Then, the preferred embodiment is to estimate the component band BDk (k is a positive integer where at least kxe2x89xa0j) by estimating band parameter values on BDk other than BDj among the component bands contained in the spectral profile of the analyzed object based on the said characteristic information by obtaining the specific characteristic information based on the said two-dimensional derivative plot with BDi removed.
Thus, the present invention is a procedure of estimating other component bands in order as for component band contained in the spectral profile of the analyzed object, using the specific component band or specific component bands already estimated by iterating estimation of component bands, creating the two-dimensional derivative plot with information on an estimated component band or estimated component band removed from the information on the analyzed object, acquiring the specific information based on this two-dimensional derivative plot and operating the estimation of other component bands based on this characteristic information.
In the two-dimensional derivative plot on spectral data of the analyzed object, the characteristic information on dominant component bands can be distinguished to some extent, however, as for characteristic information on unresolved component bands buried under the dominant component band, since it is subtle or since it overlaps with the characteristic information of other component bands overlapping with the said component band, the characteristic information of the said component band of its own becomes unclear and it is difficult to be distinguished.
Particularly, in the case where the overlapping component bands are dominant ones, the characteristic information on the unresolved component bands buried under the dominant component band can seldom be distinguished.
However, as the analytic method of the present invention shows, when the two-dimensional derivative plot is created by removing information on dominant component bands estimated from the characteristic information of the two-dimensional derivative plot of the analyzed object from the information about the analyzed object and by obtaining the two-dimensional derivative plot based on the spectral profile after removal of information on dominant component bands, the characteristic information of component bands hidden by overlapping dominant component bands appears, which enables to obtain the characteristic information about component bands hidden by overlapping, and further enables to obtain the characteristic information which has never been achieved by the two-dimensional derivative plot on the spectral data of the analyzed object.
Then, by iterating the operations of removing the profile of single or several component bands already estimated from the information about the analyzed object and by obtaining the characteristic information using the two-dimensional derivative plot based on the removal thereafter estimating other component bands, estimating the component band contained in the spectral profile of the analyzed object in order with suitability can be achieved.
Thus, the method of estimating a single component band or several component bands contained in the spectral profile of the analyzed object and finding the other component bands in order by using the two-dimensional derivative plot obtained by removing the profile of the single component band or several component bands from the spectral profile of the analyzed object, or this procedure is called xe2x80x9cBand Strippingxe2x80x9d.
Therefore, when the method of the present invention is employed, the component band that is contained in the spectral profile of the analyzed object can be estimated easily.
Further, in the preferred embodiment of the present invention, the component band is a Gaussian band, a Lorentzian band, or the mixture thereof.
In the present invention, n is preferably 1 and/or 3 and m is n+1. When the combination of n and m is (n,m)=(1,2), (3,4), the characteristic information on the spectral data appears clearly. Therefore, when the two-dimensional derivative plot is created with the combination of (n,m)=(1,2)and/or (3,4), the characteristic information can be obtained easily.
According to the method of spectrum analysis in two-dimensional representation, in the two-dimensional derivative plot where pairs of the first and second derivatives are represented in the X-Y coordinate system, when a typical local minimum indicates the existence of a corresponding component band, an X position of the said local minimum is a first approximation of band center position Xc of the said component band, setting several points on the said two-dimensional derivative plot in the vicinity of Pd, point of intersection of the said two-dimensional derivative plot with the X-axis, as candidates for the inflection point of the said component band, estimating the bandwidth of the said component band from the candidate of the said inflection point by the following Equation (1), estimating the peak height of the said component band from the distances between the said local minimum and the point(s) of intersection of vertical line passing through the said local minimum and the horizontal line(s) passing through the said candidate points, obtaining the candidates for band parameter values of the said component band, and further obtaining the constraint conditions subjected to the band parameter values for the said component band from the said two dimensional derivative plot, the relation between the bandwidth bw and the X-position of the inflection point Xp of a single band can be preferably expressed by
bw=(1/Kp) |Xcxe2x88x92XP|xe2x80x83xe2x80x83(1) 
(In Equation, bw is an estimated value of the bandwidth of a Gaussian or a Lorentzian band, where the coefficient KP is 0.42466 for Gaussian and 0.288675 for Lorentzian.)
According to the method of spectrum analysis in two-dimensional representation, in the two-dimensional derivative plot where pairs of the third and fourth derivatives are represented in the X-Y coordinate system, when a typical local maximum indicates the existence of a corresponding component band, an X position of the said local maximum is a first approximation of band center position Xc of the said component band, setting several points on the said two-dimensional derivative plot in the vicinity of Qd, point of intersection of the said two-dimensional derivative plot with the X-axis, as candidates for the secondary inflection point of the said component band, estimating the bandwidth of the said component band from the candidate of the said secondary inflection point by the following Equation (2), estimating the peak height of the said component band from the distances between the said local maximum and the point(s) of intersection of vertical line passing through the said local maximum and the horizontal line(s) passing through the said candidate points, obtaining the candidates for band parameter values of the said component band, and further obtaining the constraint conditions subjected to the band parameter values for the said component band from the said two dimensional derivative plot, the relation between the bandwidth bw and the X-position of the secondary inflection point XQ of a single band can be preferably expressed by
bw=(1/Kp) |Xcxe2x88x92XQ|xe2x80x83xe2x80x83(2) 
(In the Equation, bw is an estimated value of the bandwidth of a Gaussian or a Lorentzian band, where the coefficient KQ is 0.31508 for Gaussian and 0.16426 for Lorentzian.)
In the present invention, the method of spectral analysis in two-dimensional representation which adjusts the already estimated band parameter values can preferably be used so that the specific component band already estimated and the complementary estimated component band with all the estimated component bands other than the said estimated specific component band removed from the spectral profile of the analyzed object or two-dimensional derivative plot coincide.
With the said band stripping method, the parameter values of the each component band cannot be truly estimated and it sometimes fails into a so-called local optimum. This is because the adjacent component bands overlap and when the parameter values of the specific component band are too large or too small, the overestimation and/or underestimation affect(s) the parameter values such as band center position, bandwidth, and peak height of the adjacent component bands. Therefore, the present invention is capable of estimating the band parameter values for the component band with further suitability by introducing xe2x80x9ccomplementary estimated component bandxe2x80x9d.
The example of the coincidence between the specific estimated component band (denoted as eBD) and its complementary estimated component band (cBD) includes a method of minimizing the total sum of the distance of the iso-wavenumber lines between eBD and cBD in the two-dimensional derivative plots. By adjusting the parameter values for the specific estimated component band and by minimizing the difference between eBD and cBD, the estimated parameter values become as close to the true value as possible. In addition, by improving the degree of symmetry of the complementary estimated component band, the adjacent band parameter values are optimized. In other words, by adjusting the band parameter values for the specific estimated component band, the degree of symmetry of the complementary estimated component band is improved. Thus, introducing the complementary estimated component band and aiming to optimize the estimated band parameter values already obtained is called xe2x80x9cComplementary Matchingxe2x80x9d method.
The object of the present invention is to provide a method of spectrum analysis that can easily estimate its several component bands of the analyzed object having spectral profile containing several component bands. In addition, it can be applicable to spectral data such as infrared spectra, visible light spectra, ultraviolet spectra, Raman spectra, X-ray diffractogram, and chromatogram, and the like.